# WhiteNoiseProcess

WhiteNoiseProcess[]

represents a Gaussian white noise process with mean 0 and standard deviation 1.

WhiteNoiseProcess[σ]

represents a Gaussian white noise process with mean 0 and standard deviation σ.

WhiteNoiseProcess[dist]

represents a white noise process based on the distribution dist.

# Details

• WhiteNoiseProcess is also known as independent identically distributed (iid) process.
• WhiteNoiseProcess is a discrete-time random process.
• The slices of WhiteNoiseProcess are assumed to be independent and identically distributed random variables.
• The distribution dist can be any univariate distribution with mean 0 and finite variance.
• WhiteNoiseProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.

# Examples

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## Basic Examples(1)

Define a Gaussian white noise process:

Simulate the process:

Mean and variance functions:

Covariance function:

## Scope(2)

Simulate a white noise process having normal slices with standard deviation 5:

Uniform distribution slices:

Discrete uniform distribution slices:

Process parameter estimation:

Estimate the process parameters from sample data:

## Applications(2)

Add white noise to a periodic signal:

Define a moving-average process:

Simulate the process:

Mean, variance, and kurtosis for the process:

Compare with the property values for the corresponding MAProcess:

## Properties & Relations(6)

WhiteNoiseProcess is a discrete-time process:

The states may either be continuous or discrete:

SliceDistribution[WhiteNoiseProcess[dist],t] is equal to dist:

Multivariate slices are products of dist with itself:

The slice mean is always zero:

WhiteNoiseProcess is uncorrelated according to the AutocorrelationTest:

Gaussian white noise is a special case of an MAProcess:

Compare covariance functions:

## Possible Issues(1)

EstimatedProcess fails for this example involving white noise from a uniform distribution:

Using a symmetric interval for UniformDistribution helps in this case:

## Neat Examples(1)

A family of white noise processes:

Wolfram Research (2014), WhiteNoiseProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html.

#### Text

Wolfram Research (2014), WhiteNoiseProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html.

#### CMS

Wolfram Language. 2014. "WhiteNoiseProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html.

#### APA

Wolfram Language. (2014). WhiteNoiseProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html

#### BibTeX

@misc{reference.wolfram_2024_whitenoiseprocess, author="Wolfram Research", title="{WhiteNoiseProcess}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html}", note=[Accessed: 29-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_whitenoiseprocess, organization={Wolfram Research}, title={WhiteNoiseProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/WhiteNoiseProcess.html}, note=[Accessed: 29-May-2024 ]}